Spanmatrix Problem

_Background info to Spanmatrix

| Spanmatrix 'S' is an 3x3 matrix that describe normal stress 'σ' and shear stresses 'τ' in one plane with the coordinate plane, through a body, that affected by internal and external forces. Eigenvectors to the matrix 'S' is "principal stress direction arena", corresponding eigenvalues the normal stress is in plane perpendicular to eigenvector.

In these planes is 'τ'=0. The matrix 'S' is always symmetric, and therefore always diagonalizable. Spanvector 's' on a special cut surface with unit normal vector 'n' is given by s=Sn.

Here is the the formula:

1) σ=n^T*Sn

If 'n' is normalized eigenvector to 'S', then s=Sn=λn, there λ is corresponding eigenvector. In this case is:

σ = IIsII = λ och τ = 0. Thereof we have the primary span and the principal stress direction.|

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(Most of it above it is just not so important as it looks like.)

The stress state in a point Q in a body have been calculated with the finite element method and is expressed in a Cartesian coordinate system (x, y, z) spanmatrix

S=[30 0 10; 0 30 10; 10 10 30] Mpa (I think this is Mega Pascal)

Count normal stress 'σ' and shear stresses 'τ' on a cut surface with unit normal vector 'n', there n=1/sqrt(5) * [1,2,0]^T.

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My suggestions

According to 1), there n^T, we have to first multiple n with the 1x3 matrix and then transpose the whole thing.

n=[1/sqrt(5), 2/sqrt(5), 0]^T

n^T=[[1/sqrt(5); 2/sqrt(5); 0]

And now, it would be suitable to count σ with forumla 1) σ=n^T*Sn. But it didn't work form me. Matlab says the dimensions are wrong. So what should I do here?

Regards Regina

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Answer 1

[Walter Roberson]

The "^" operator can only be used with square matrices, not with vectors, as the "^" operator is matrix power. For example, A^2 is A*A which is matrix multiplication.

Perhaps you need the .^ operator.